What are the integrals defined for $\mathbb{R}^n$?

In multivariate calculus,

1. I was wondering what types of integrals are studied? Here are my naive view:

   • Multiple integral: If I understand correctly, it is just a plain generalization of Riemann integral on $\mathbb{R}$ to on $\mathbb{R^n}$.
   • Integral of differential forms: It is something I am not able to truly understand. Is it a special kind of Lebesgue integral? Does its definition rely on measure?

2. When trying to compare them together, I have some further questions:

   1. Are multiple integrals and integrals of differential forms two different types of integrals? Do they belong to some common type of integral, similarly to that Riemann integral and Lebesgue-Stieltjes integrals both belong to Lebesgue integral? How are they related?

   2. Is it correct that multiple integrals have no orientation involved, but an integral of differential forms does, in the sense of changing the order of dummy variables in $dx_1 dx_2$ will or will not change the integral?

3. What type of integral is used in vector calculus, for topics such as gradient, divergence, curl, Laplacian, the gradient theorem, Green's theorem, Stokes' theorem, divergence theorem? Are the line integral, surface integral and volume integral defined as belonging to Lebesgue integrals or Riemann integrals, integrals of differential forms, or something else? Do their definitions rely on measure?

Thanks and regards!

Solution 1:

I am reading this question as saying: There are too many integrals and that confuses me. Here is my attempt to clear the waters a little.

First, let's consider our old friend the Riemann integral. We used this to find the areas between curves (or volume bounded by surfaces if we go up a dimension). In fact, The Riemann integral was presented as the solution to this problem, and it works very well except that some things which are "obviously true" fail to even make sense for this integral. The example I have in mind is that $\int_{\mathbb{R}}\chi_{\mathbb{Q}} = 0$, where $\chi_{\mathbb{Q}}$ is the indicator function of the rationals, is something that we want to be true, but the function fails to be Riemann integrable. The Lebesgue integral is the solution to this problem.

Even though you have to learn a whole new theory of Lebesgue integration on measure spaces you can think of it as a patch which makes the dominated convergence theorem true since (as long as you pick the right definition of measurable function) the two integrals agree whenever the Riemann integral exists. This is done by (more or less) slicing horizontally rather that vertically, and having a more robust version of "area".

Your questions about differential forms are possibly answered by the following statement:

Calculus is awesome!

What I mean is that we have learned a tremendous amount by doing calculus, and wouldn't it be great if we could use this tool on "shapes" other than (open subsets of) $\mathbb{R}^n$, say for example curves and surfaces. As it turns out differential forms are the solution to this problem. Rather than go into a whole course about manifolds let me just say that whatever a manifold is, $\mathbb{R}^n$ is one and the forms notion of integration coincides with the regular way. So we can think of differential forms less like a new way to integrate and more like a way to extend the value of integration while at the same time giving a new language in which to reinterpret what what we know in $\mathbb{R}^n$.

To sum up when you are integrating you are always using differential forms and the Lebesgue integral, but of course you don't need to know that most of the time.

I hope this answers at least a few of your questions.